Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
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A group homomorphism is injective if and only if its kernel is trivial
Statement
A group homomorphism is injective if and only if its kernel is trivial.
For a group homomorphism , is injective exactly when .
Facts & Assumptions
Given: A group homomorphism .
Homomorphisms preserve inverses and products (A group homomorphism automatically satisfies and , and for every ; for monoid homomorphisms preservation of the identity must be assumed).
Injectivity means that equal values have equal arguments (Injection, surjection, bijection).
In a group, implies (In a group , and , the order of the last product being essential).
Proof
If is injective and , then , so .
Conversely, if and , then , whence and .
The two implications prove the equivalence.
Depends on
- The kernel and image of a group homomorphism
- A group homomorphism automatically satisfies $f(e) = e'$ and $f(g^{-1}) = f(g)^{-1}$, and $f(g^{n}) = f(g)^{n}$ for every $n \in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed
- Injection, surjection, bijection
- In a group $e^{-1} = e$, $(g^{-1})^{-1} = g$ and $(gh)^{-1} = h^{-1}g^{-1}$, the order of the last product being essential
Used by
- Every canonical factor map into a free product is injective Corollary
- The factor maps into a free product with amalgamation are injective Corollary
- The doubling endomorphism of (ℤ,+) has trivial kernel but is not surjective Counterexample
- Ascending HNN extensions of injective endomorphisms Definition
- Free products with amalgamation along monomorphisms Definition
- For primes p<q, nontrivial actions of Cₚ on C_q exist exactly when p∣(q-1) and are unique up to automorphisms Lemma
- The edge-group presentation is equivalent to the associated-subgroup presentation Lemma
- Cayley's theorem: every group G is isomorphic to a subgroup of Sym(G) Theorem
- First isomorphism theorem for groups: G/ker f congimf Theorem
- Internal direct products are external direct products, equivalently every element has a unique factorisation Theorem
- Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel Theorem
- Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent Theorem
Dependency tree · two levels
22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)